Finite Group DatabasebetaFinite Group Databasebeta| order | 1680 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Hall | C3×C5×C7 ⋊ #41 |
| Hall | C5×C7 ⋊ #237 |
| Hall | C3×C7 ⋊ #711 |
| Hall | C3×C5 ⋊ #1159 |
| Sylow | C7 ⋊ #6819 |
| Sylow | C5 ⋊ #66684 |
| Sylow | C3 ⋊ #10587953 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C2×D7×D15 |
| commutator, socle, cosocle | C2×C3×C5×C7 | C23 |
| Fitting | C22×C3×C5×C7 | C22 |
| hypercenter | C22 | D7×D15 |
| hypocenter | C3×C5×C7 | #41 |
| LCS | G ▷ C2×C3×C5×C7 ▷ C3×C5×C7 | |
| UCS | C1 ◁ C2 ◁ C22 | |
160 conjugacy classes of subgroups; only the 50 normal subgroups are shown.
This group has 160 conjugacy classes of subgroups — too many to draw. Only its 50 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 840 |
Polycyclic presentation:
2^4,3,5,7|b^2=d,c^a=cd,g^a=g^6,c^b=cd,e^b=e^2,f^b=f^4144 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1680, 721);